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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.root_comparison.root_n_comparison"></a><a class="link" href="root_n_comparison.html" title="Comparison of Nth-root Finding Algorithms">Comparison
      of Nth-root Finding Algorithms</a>
</h3></div></div></div>
<p>
        A second example compares four generalized nth-root finding algorithms for
        various n-th roots (5, 7 and 13) of a single value 28.0, for four floating-point
        types, <code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code>, <code class="computeroutput"><span class="keyword">long</span>
        <span class="keyword">double</span></code> and a <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
        type <code class="computeroutput"><span class="identifier">cpp_bin_float_50</span></code>. In
        each case the target accuracy was set using our "recommended" accuracy
        limits (or at least limits that make a good starting point - which is likely
        to give close to full accuracy without resorting to unnecessary iterations).
      </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                <p>
                  Function
                </p>
              </th>
<th>
                <p>
                  Precision Requested
                </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  TOMS748
                </p>
              </td>
<td>
                <p>
                  numeric_limits&lt;T&gt;::digits - 2
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Newton
                </p>
              </td>
<td>
                <p>
                  floor(numeric_limits&lt;T&gt;::digits * 0.6)
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Halley
                </p>
              </td>
<td>
                <p>
                  floor(numeric_limits&lt;T&gt;::digits * 0.4)
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Schröder
                </p>
              </td>
<td>
                <p>
                  floor(numeric_limits&lt;T&gt;::digits * 0.4)
                </p>
              </td>
</tr>
</tbody>
</table></div>
<p>
        Tests used Microsoft Visual Studio 2013 (Update 1) and GCC 4.9.1 using source
        code <a href="../../../../example/root_n_finding_algorithms.cpp" target="_top">root_n_finding_algorithms.cpp</a>.
      </p>
<p>
        The timing uncertainty (especially using MSVC) is at least 5% of normalized
        time 'Norm'.
      </p>
<p>
        To pick out the 'best' and 'worst' algorithms are highlighted in blue and
        red. More than one result can be 'best' when normalized times are indistinguishable
        within the uncertainty.
      </p>
<h4>
<a name="math_toolkit.root_comparison.root_n_comparison.h0"></a>
        <span class="phrase"><a name="math_toolkit.root_comparison.root_n_comparison.program_example_root_n_finding_a"></a></span><a class="link" href="root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.program_example_root_n_finding_a">Program
        ..\example\root_n_finding_algorithms.cpp, Microsoft Visual C++ version 14.1,
        Dinkumware standard library version 650, Win32 Compiled in optimise mode.,
        _X86_SSE2</a>
      </h4>
<p>
        Fraction of full accuracy 1
      </p>
<div class="table">
<a name="math_toolkit.root_comparison.root_n_comparison.root_5"></a><p class="title"><b>Table 10.3. 5th root(28) for float, double, long double and cpp_bin_float_50 types,
        using _X86_SSE2</b></p>
<div class="table-contents"><table class="table" summary="5th root(28) for float, double, long double and cpp_bin_float_50 types,
        using _X86_SSE2">
<colgroup>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  float
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  double
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  long d
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  cpp50
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<td class="auto-generated"> </td>
<td class="auto-generated"> </td>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  Algo
                </p>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  TOMS748
                </p>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  457
                </p>
              </td>
<td>
                <p>
                  2.00
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  11
                </p>
              </td>
<td>
                <p>
                  860
                </p>
              </td>
<td>
                <p>
                  3.54
                </p>
              </td>
<td>
                <p>
                  1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  11
                </p>
              </td>
<td>
                <p>
                  806
                </p>
              </td>
<td>
                <p>
                  3.02
                </p>
              </td>
<td>
                <p>
                  1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  12
                </p>
              </td>
<td>
                <p>
                  226875
                </p>
              </td>
<td>
                <p>
                  <span class="red">8.11</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Newton
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  228
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  243
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  -1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  298
                </p>
              </td>
<td>
                <p>
                  1.12
                </p>
              </td>
<td>
                <p>
                  -1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  27968
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Halley
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
                <p>
                  250
                </p>
              </td>
<td>
                <p>
                  1.10
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  268
                </p>
              </td>
<td>
                <p>
                  1.10
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  267
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  52812
                </p>
              </td>
<td>
                <p>
                  1.89
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Schröder
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
                <p>
                  256
                </p>
              </td>
<td>
                <p>
                  1.12
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  271
                </p>
              </td>
<td>
                <p>
                  1.12
                </p>
              </td>
<td>
                <p>
                  -1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  270
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.01</span>
                </p>
              </td>
<td>
                <p>
                  -1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  61406
                </p>
              </td>
<td>
                <p>
                  2.20
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.root_comparison.root_n_comparison.root_7"></a><p class="title"><b>Table 10.4. 7th root(28) for float, double, long double and cpp_bin_float_50 types,
        using _X86_SSE2</b></p>
<div class="table-contents"><table class="table" summary="7th root(28) for float, double, long double and cpp_bin_float_50 types,
        using _X86_SSE2">
<colgroup>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  float
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  double
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  long d
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  cpp50
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<td class="auto-generated"> </td>
<td class="auto-generated"> </td>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  Algo
                </p>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  TOMS748
                </p>
              </td>
<td>
                <p>
                  12
                </p>
              </td>
<td>
                <p>
                  825
                </p>
              </td>
<td>
                <p>
                  3.06
                </p>
              </td>
<td>
                <p>
                  1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  15
                </p>
              </td>
<td>
                <p>
                  1145
                </p>
              </td>
<td>
                <p>
                  <span class="red">4.06</span>
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  15
                </p>
              </td>
<td>
                <p>
                  1159
                </p>
              </td>
<td>
                <p>
                  <span class="red">4.17</span>
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  14
                </p>
              </td>
<td>
                <p>
                  295781
                </p>
              </td>
<td>
                <p>
                  <span class="red">8.12</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Newton
                </p>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  270
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  282
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  278
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  8
                </p>
              </td>
<td>
                <p>
                  36406
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Halley
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  303
                </p>
              </td>
<td>
                <p>
                  1.12
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  329
                </p>
              </td>
<td>
                <p>
                  1.17
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  335
                </p>
              </td>
<td>
                <p>
                  1.21
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  78281
                </p>
              </td>
<td>
                <p>
                  2.15
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Schröder
                </p>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  340
                </p>
              </td>
<td>
                <p>
                  1.26
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  432
                </p>
              </td>
<td>
                <p>
                  1.53
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  367
                </p>
              </td>
<td>
                <p>
                  1.32
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  85156
                </p>
              </td>
<td>
                <p>
                  2.34
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.root_comparison.root_n_comparison.root_11"></a><p class="title"><b>Table 10.5. 11th root(28) for float, double, long double and cpp_bin_float_50
        types, using _X86_SSE2</b></p>
<div class="table-contents"><table class="table" summary="11th root(28) for float, double, long double and cpp_bin_float_50
        types, using _X86_SSE2">
<colgroup>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  float
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  double
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  long d
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  cpp50
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<td class="auto-generated"> </td>
<td class="auto-generated"> </td>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  Algo
                </p>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  TOMS748
                </p>
              </td>
<td>
                <p>
                  12
                </p>
              </td>
<td>
                <p>
                  714
                </p>
              </td>
<td>
                <p>
                  3.16
                </p>
              </td>
<td>
                <p>
                  -2
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  14
                </p>
              </td>
<td>
                <p>
                  909
                </p>
              </td>
<td>
                <p>
                  <span class="red">4.19</span>
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  14
                </p>
              </td>
<td>
                <p>
                  793
                </p>
              </td>
<td>
                <p>
                  3.69
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  17
                </p>
              </td>
<td>
                <p>
                  211718
                </p>
              </td>
<td>
                <p>
                  <span class="red">9.28</span>
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Newton
                </p>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  226
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  217
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  215
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  9
                </p>
              </td>
<td>
                <p>
                  22812
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Halley
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  262
                </p>
              </td>
<td>
                <p>
                  1.16
                </p>
              </td>
<td>
                <p>
                  -1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  260
                </p>
              </td>
<td>
                <p>
                  1.20
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  260
                </p>
              </td>
<td>
                <p>
                  1.21
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  40781
                </p>
              </td>
<td>
                <p>
                  1.79
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Schröder
                </p>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  332
                </p>
              </td>
<td>
                <p>
                  1.47
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  314
                </p>
              </td>
<td>
                <p>
                  1.45
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  310
                </p>
              </td>
<td>
                <p>
                  1.44
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  8
                </p>
              </td>
<td>
                <p>
                  67187
                </p>
              </td>
<td>
                <p>
                  2.95
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><h4>
<a name="math_toolkit.root_comparison.root_n_comparison.h1"></a>
        <span class="phrase"><a name="math_toolkit.root_comparison.root_n_comparison.program_root_n_finding_algorithm"></a></span><a class="link" href="root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.program_root_n_finding_algorithm">Program
        root_n_finding_algorithms.cpp, Microsoft Visual C++ version 12.0, Dinkumware
        standard library version 610, Win32 Compiled in optimise mode., _X64_AVX</a>
      </h4>
<p>
        Fraction of full accuracy 1
      </p>
<div class="table">
<a name="math_toolkit.root_comparison.root_n_comparison.root_5_0"></a><p class="title"><b>Table 10.6. 5th root(28) for float, double, long double and cpp_bin_float_50 types,
        using _X64_AVX</b></p>
<div class="table-contents"><table class="table" summary="5th root(28) for float, double, long double and cpp_bin_float_50 types,
        using _X64_AVX">
<colgroup>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  float
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  double
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  long d
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  cpp50
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<td class="auto-generated"> </td>
<td class="auto-generated"> </td>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  Algo
                </p>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  TOMS748
                </p>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  239
                </p>
              </td>
<td>
                <p>
                  1.50
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  11
                </p>
              </td>
<td>
                <p>
                  451
                </p>
              </td>
<td>
                <p>
                  2.53
                </p>
              </td>
<td>
                <p>
                  1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  11
                </p>
              </td>
<td>
                <p>
                  439
                </p>
              </td>
<td>
                <p>
                  2.49
                </p>
              </td>
<td>
                <p>
                  1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  12
                </p>
              </td>
<td>
                <p>
                  90312
                </p>
              </td>
<td>
                <p>
                  <span class="red">7.51</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Newton
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  159
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  178
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  -1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  176
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  -1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  12031
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Halley
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
                <p>
                  168
                </p>
              </td>
<td>
                <p>
                  1.06
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  203
                </p>
              </td>
<td>
                <p>
                  1.14
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  198
                </p>
              </td>
<td>
                <p>
                  1.13
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  20937
                </p>
              </td>
<td>
                <p>
                  1.74
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Schröder
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
                <p>
                  173
                </p>
              </td>
<td>
                <p>
                  1.09
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  206
                </p>
              </td>
<td>
                <p>
                  1.16
                </p>
              </td>
<td>
                <p>
                  -1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  203
                </p>
              </td>
<td>
                <p>
                  1.15
                </p>
              </td>
<td>
                <p>
                  -1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  26250
                </p>
              </td>
<td>
                <p>
                  2.18
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.root_comparison.root_n_comparison.root_7_0"></a><p class="title"><b>Table 10.7. 7th root(28) for float, double, long double and cpp_bin_float_50 types,
        using _X64_AVX</b></p>
<div class="table-contents"><table class="table" summary="7th root(28) for float, double, long double and cpp_bin_float_50 types,
        using _X64_AVX">
<colgroup>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  float
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  double
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  long d
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  cpp50
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<td class="auto-generated"> </td>
<td class="auto-generated"> </td>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  Algo
                </p>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  TOMS748
                </p>
              </td>
<td>
                <p>
                  12
                </p>
              </td>
<td>
                <p>
                  385
                </p>
              </td>
<td>
                <p>
                  2.19
                </p>
              </td>
<td>
                <p>
                  1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  15
                </p>
              </td>
<td>
                <p>
                  635
                </p>
              </td>
<td>
                <p>
                  3.13
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  15
                </p>
              </td>
<td>
                <p>
                  621
                </p>
              </td>
<td>
                <p>
                  3.17
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  14
                </p>
              </td>
<td>
                <p>
                  114843
                </p>
              </td>
<td>
                <p>
                  <span class="red">6.81</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Newton
                </p>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  176
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  203
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  196
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  8
                </p>
              </td>
<td>
                <p>
                  16875
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Halley
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  209
                </p>
              </td>
<td>
                <p>
                  1.19
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  254
                </p>
              </td>
<td>
                <p>
                  1.25
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  246
                </p>
              </td>
<td>
                <p>
                  1.26
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  32343
                </p>
              </td>
<td>
                <p>
                  1.92
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Schröder
                </p>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  223
                </p>
              </td>
<td>
                <p>
                  1.27
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  273
                </p>
              </td>
<td>
                <p>
                  1.34
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  275
                </p>
              </td>
<td>
                <p>
                  1.40
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  45156
                </p>
              </td>
<td>
                <p>
                  2.68
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.root_comparison.root_n_comparison.root_11_0"></a><p class="title"><b>Table 10.8. 11th root(28) for float, double, long double and cpp_bin_float_50
        types, using _X64_AVX</b></p>
<div class="table-contents"><table class="table" summary="11th root(28) for float, double, long double and cpp_bin_float_50
        types, using _X64_AVX">
<colgroup>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  float
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  double
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  long d
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  cpp50
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<td class="auto-generated"> </td>
<td class="auto-generated"> </td>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  Algo
                </p>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  TOMS748
                </p>
              </td>
<td>
                <p>
                  12
                </p>
              </td>
<td>
                <p>
                  467
                </p>
              </td>
<td>
                <p>
                  2.42
                </p>
              </td>
<td>
                <p>
                  -2
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  14
                </p>
              </td>
<td>
                <p>
                  648
                </p>
              </td>
<td>
                <p>
                  3.06
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  14
                </p>
              </td>
<td>
                <p>
                  640
                </p>
              </td>
<td>
                <p>
                  2.99
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  17
                </p>
              </td>
<td>
                <p>
                  170000
                </p>
              </td>
<td>
                <p>
                  <span class="red">8.85</span>
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Newton
                </p>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  193
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  212
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  214
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  9
                </p>
              </td>
<td>
                <p>
                  19218
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Halley
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  209
                </p>
              </td>
<td>
                <p>
                  1.08
                </p>
              </td>
<td>
                <p>
                  -1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  256
                </p>
              </td>
<td>
                <p>
                  1.21
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  250
                </p>
              </td>
<td>
                <p>
                  1.17
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  32656
                </p>
              </td>
<td>
                <p>
                  1.70
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Schröder
                </p>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  248
                </p>
              </td>
<td>
                <p>
                  1.28
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  306
                </p>
              </td>
<td>
                <p>
                  1.44
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  298
                </p>
              </td>
<td>
                <p>
                  1.39
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  8
                </p>
              </td>
<td>
                <p>
                  53437
                </p>
              </td>
<td>
                <p>
                  2.78
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><h4>
<a name="math_toolkit.root_comparison.root_n_comparison.h2"></a>
        <span class="phrase"><a name="math_toolkit.root_comparison.root_n_comparison.program_example_root_n_finding_0"></a></span><a class="link" href="root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.program_example_root_n_finding_0">Program
        ..\example\root_n_finding_algorithms.cpp, GNU C++ version 7.1.0, GNU libstdc++
        version 20170502, Win32 Compiled in optimise mode., _X64_SSE2</a>
      </h4>
<p>
        Fraction of full accuracy 1
      </p>
<div class="table">
<a name="math_toolkit.root_comparison.root_n_comparison.root_5_1"></a><p class="title"><b>Table 10.9. 5th root(28) for float, double, long double and cpp_bin_float_50 types,
        using _X64_SSE2</b></p>
<div class="table-contents"><table class="table" summary="5th root(28) for float, double, long double and cpp_bin_float_50 types,
        using _X64_SSE2">
<colgroup>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  float
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  double
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  long d
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  cpp50
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<td class="auto-generated"> </td>
<td class="auto-generated"> </td>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  Algo
                </p>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  TOMS748
                </p>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  206
                </p>
              </td>
<td>
                <p>
                  2.24
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  11
                </p>
              </td>
<td>
                <p>
                  460
                </p>
              </td>
<td>
                <p>
                  <span class="red">4.04</span>
                </p>
              </td>
<td>
                <p>
                  1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  9
                </p>
              </td>
<td>
                <p>
                  554
                </p>
              </td>
<td>
                <p>
                  <span class="red">4.40</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  12
                </p>
              </td>
<td>
                <p>
                  57656
                </p>
              </td>
<td>
                <p>
                  <span class="red">8.39</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Newton
                </p>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  92
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  114
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  -1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  126
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  6875
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Halley
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
                <p>
                  106
                </p>
              </td>
<td>
                <p>
                  1.15
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  134
                </p>
              </td>
<td>
                <p>
                  1.18
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  178
                </p>
              </td>
<td>
                <p>
                  1.41
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  12500
                </p>
              </td>
<td>
                <p>
                  1.82
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Schröder
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
                <p>
                  126
                </p>
              </td>
<td>
                <p>
                  1.37
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  143
                </p>
              </td>
<td>
                <p>
                  1.25
                </p>
              </td>
<td>
                <p>
                  -1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  3
                </p>
              </td>
<td>
                <p>
                  198
                </p>
              </td>
<td>
                <p>
                  1.57
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  15312
                </p>
              </td>
<td>
                <p>
                  2.23
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.root_comparison.root_n_comparison.root_7_1"></a><p class="title"><b>Table 10.10. 7th root(28) for float, double, long double and cpp_bin_float_50 types,
        using _X64_SSE2</b></p>
<div class="table-contents"><table class="table" summary="7th root(28) for float, double, long double and cpp_bin_float_50 types,
        using _X64_SSE2">
<colgroup>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  float
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  double
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  long d
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  cpp50
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<td class="auto-generated"> </td>
<td class="auto-generated"> </td>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  Algo
                </p>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  TOMS748
                </p>
              </td>
<td>
                <p>
                  12
                </p>
              </td>
<td>
                <p>
                  345
                </p>
              </td>
<td>
                <p>
                  2.09
                </p>
              </td>
<td>
                <p>
                  1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  15
                </p>
              </td>
<td>
                <p>
                  615
                </p>
              </td>
<td>
                <p>
                  3.14
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  13
                </p>
              </td>
<td>
                <p>
                  875
                </p>
              </td>
<td>
                <p>
                  3.98
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  14
                </p>
              </td>
<td>
                <p>
                  70937
                </p>
              </td>
<td>
                <p>
                  <span class="red">7.32</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Newton
                </p>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  165
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  196
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  220
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  8
                </p>
              </td>
<td>
                <p>
                  9687
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Halley
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  193
                </p>
              </td>
<td>
                <p>
                  1.17
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  239
                </p>
              </td>
<td>
                <p>
                  1.22
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  298
                </p>
              </td>
<td>
                <p>
                  1.35
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  19062
                </p>
              </td>
<td>
                <p>
                  1.97
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Schröder
                </p>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  217
                </p>
              </td>
<td>
                <p>
                  1.32
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  270
                </p>
              </td>
<td>
                <p>
                  1.38
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  367
                </p>
              </td>
<td>
                <p>
                  1.67
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  27343
                </p>
              </td>
<td>
                <p>
                  2.82
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.root_comparison.root_n_comparison.root_11_1"></a><p class="title"><b>Table 10.11. 11th root(28) for float, double, long double and cpp_bin_float_50
        types, using _X64_SSE2</b></p>
<div class="table-contents"><table class="table" summary="11th root(28) for float, double, long double and cpp_bin_float_50
        types, using _X64_SSE2">
<colgroup>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  float
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  double
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  long d
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
              </th>
<th>
                <p>
                  cpp50
                </p>
              </th>
<th>
              </th>
<th>
              </th>
<td class="auto-generated"> </td>
<td class="auto-generated"> </td>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  Algo
                </p>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  Its
                </p>
              </td>
<td>
                <p>
                  Times
                </p>
              </td>
<td>
                <p>
                  Norm
                </p>
              </td>
<td>
                <p>
                  Dis
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  TOMS748
                </p>
              </td>
<td>
                <p>
                  12
                </p>
              </td>
<td>
                <p>
                  412
                </p>
              </td>
<td>
                <p>
                  2.15
                </p>
              </td>
<td>
                <p>
                  -2
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  14
                </p>
              </td>
<td>
                <p>
                  646
                </p>
              </td>
<td>
                <p>
                  2.96
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  14
                </p>
              </td>
<td>
                <p>
                  1054
                </p>
              </td>
<td>
                <p>
                  <span class="red">4.22</span>
                </p>
              </td>
<td>
                <p>
                  1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  17
                </p>
              </td>
<td>
                <p>
                  107187
                </p>
              </td>
<td>
                <p>
                  <span class="red">9.53</span>
                </p>
              </td>
<td>
                <p>
                  2
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Newton
                </p>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  192
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  218
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  250
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  9
                </p>
              </td>
<td>
                <p>
                  11250
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.00</span>
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Halley
                </p>
              </td>
<td>
                <p>
                  4
                </p>
              </td>
<td>
                <p>
                  200
                </p>
              </td>
<td>
                <p>
                  <span class="blue">1.04</span>
                </p>
              </td>
<td>
                <p>
                  -1
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  243
                </p>
              </td>
<td>
                <p>
                  1.11
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  5
                </p>
              </td>
<td>
                <p>
                  345
                </p>
              </td>
<td>
                <p>
                  1.38
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  19687
                </p>
              </td>
<td>
                <p>
                  1.75
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
<tr>
<td>
                <p>
                  Schröder
                </p>
              </td>
<td>
                <p>
                  6
                </p>
              </td>
<td>
                <p>
                  254
                </p>
              </td>
<td>
                <p>
                  1.32
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  321
                </p>
              </td>
<td>
                <p>
                  1.47
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  7
                </p>
              </td>
<td>
                <p>
                  471
                </p>
              </td>
<td>
                <p>
                  1.88
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  8
                </p>
              </td>
<td>
                <p>
                  33281
                </p>
              </td>
<td>
                <p>
                  2.96
                </p>
              </td>
<td>
                <p>
                  0
                </p>
              </td>
<td>
              </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><p>
        Some tentative conclusions can be drawn from this limited exercise.
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            Perhaps surprisingly, there is little difference between the various
            algorithms for <a href="http://en.cppreference.com/w/cpp/language/types" target="_top">fundamental
            (built-in) types</a> floating-point types. Using the first derivatives
            (<a class="link" href="../roots_deriv.html#math_toolkit.roots_deriv.newton">Newton-Raphson iteration</a>)
            is usually the best, but while the improvement over the no-derivative
            <a href="http://portal.acm.org/citation.cfm?id=210111" target="_top">TOMS Algorithm
            748: enclosing zeros of continuous functions</a> is considerable
            in number of iterations, but little in execution time. This reflects
            the fact that the function we are finding the root for is trivial to
            evaluate, so runtimetimes are dominated by the time taken by the boilerplate
            code in each method.
          </li>
<li class="listitem">
            The extra cost of evaluating the second derivatives (<a class="link" href="../roots_deriv.html#math_toolkit.roots_deriv.halley">Halley</a>
            or <a class="link" href="../roots_deriv.html#math_toolkit.roots_deriv.schroder">Schröder</a>)
            is usually too much for any net benefit: as with the cube root, these
            functors are so cheap to evaluate that the runtime is largely dominated
            by the complexity of the root finding method.
          </li>
<li class="listitem">
            For a <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
            floating-point type, the <a class="link" href="../roots_deriv.html#math_toolkit.roots_deriv.newton">Newton-Raphson
            iteration</a> is a clear winner with a several-fold gain over <a href="http://portal.acm.org/citation.cfm?id=210111" target="_top">TOMS Algorithm 748:
            enclosing zeros of continuous functions</a>, and again no improvement
            from the second-derivative algorithms.
          </li>
<li class="listitem">
            The run-time of 50 decimal-digit <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
            is about 30-fold greater than <code class="computeroutput"><span class="keyword">double</span></code>.
          </li>
<li class="listitem">
            The column 'dis' showing the number of bits distance from the correct
            result. The Newton-Raphson algorithm shows a bit or two better accuracy
            than <a href="http://portal.acm.org/citation.cfm?id=210111" target="_top">TOMS Algorithm
            748: enclosing zeros of continuous functions</a>.
          </li>
<li class="listitem">
            The goodness of the 'guess' is especially crucial for <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.
            Separate experiments show that evaluating the 'guess' using <code class="computeroutput"><span class="keyword">double</span></code> allows convergence to the final
            exact result in one or two iterations. So in this contrived example,
            crudely dividing the exponent by N for a 'guess', it would be far better
            to use a <code class="computeroutput"><span class="identifier">pow</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span></code>
            or , if more precise <code class="computeroutput"><span class="identifier">pow</span><span class="special">&lt;</span><span class="keyword">long</span> <span class="keyword">double</span><span class="special">&gt;</span></code>,
            function to estimate a 'guess'. The limitation of this tactic is that
            the range of possible (exponent) values may be less than the multiprecision
            type.
          </li>
<li class="listitem">
            Using floating-point extension <a href="http://en.wikipedia.org/wiki/SSE2" target="_top">SSE2
            instructions</a> made a modest ten-percent speedup.
          </li>
<li class="listitem">
            Using MSVC, there was some improvement using 64-bit, markedly for <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.
          </li>
<li class="listitem">
            The GCC compiler 4.9.1 using 64-bit was at least five-folder faster that
            32-bit, apparently reflecting better optimization.
          </li>
</ul></div>
<p>
        Clearly, your mileage <span class="bold"><strong>will vary</strong></span>, but in
        summary, <a class="link" href="../roots_deriv.html#math_toolkit.roots_deriv.newton">Newton-Raphson iteration</a>
        seems the first choice of algorithm, and effort to find a good 'guess' the
        first speed-up target, especially for <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.
        And of course, compiler optimisation is crucial for speed.
      </p>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
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